Parts of the mesh don't move when moving bones
The vertex of those parts that stay behind means you don't have them assigned to any bones to move them. Check all the vertex groups in weight painting mode and most likely they will be blue for all vertex groups. 

You can select a vertex group while the mouse pointer is still in the list you can use the up/down arrow key to select all the vertex in the list to check quicker.

Then you need to weight paint them with the Add brush to the desired vertex group.